Question

In a dataset where all values are the same, what can be said about the mean, median, and mode?

• A. They will all be different
• B. The mean will be higher than the median and mode
• C. The median will be higher than the mean and mode
• D. They will all be the same

Hint:

For any uniform dataset (e.g., \( \{5, 5, 5, 5\} \)):
- Mean = 5
- Median = 5
- Mode = 5
All measures of central tendency converge to this single value.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When all elements in a dataset are identical, the dataset has zero variance. Let us analyze how this affects the three measures of central tendency: mean, median, and mode.

Step 2: Key Formula or Approach:
Let the dataset contain \( n \) observations, where each observation is equal to a constant value \( k \):
\[ S = \{k, k, k, \dots, k\} \]

Step 3: Detailed Explanation:
Let us calculate the statistical measures for this dataset:
1. Mean:
\[ \text{Mean} = \frac{\sum_{i=1}^{n} k}{n} = \frac{n \times k}{n} = k \]
2. Median:
Since all elements are \( k \), any middle element (or the average of two middle elements) is:
\[ \text{Median} = k \]
3. Mode:
The value \( k \) appears with a relative frequency of \( 100\% \), making it the most frequent value:
\[ \text{Mode} = k \]
Because all three measures are equal to \( k \), they are identical.

Step 4: Final Answer
In a dataset with zero variance where all values are identical, the mean, median, and mode are equal.